Biophysical Journal Volume 73 September 1997 1617-1626

Electrorotation of Colloidal Particles and Cells Depends onSurface Charge

Hannes MaierSection Physiological Acoustics, Department of Otolaryngology, University of Tubingen, D-72076 Tubingen, Germany

ABSTRACT The importance of surface conductivity to the frequency-dependent polarizability and the rotation of particlesin circular electric fields (electrorotation) is emphasized by various theoretical and experimental investigations. Althoughsurface conductivity seems to be naturally related to the ionic double layer, there is rare experimental evidence of a directrelationship. To highlight the role of surface charges in electrorotation, an apparatus was developed with a symmetricalthree-electrode arrangement for field frequencies between 25 Hz and 80 MHz. The three-dimensional electrostatic fielddistribution between the electrodes was evaluated numerically. With this device, rotating, gradient, and homogeneous electricfields of defined precision and homogeneity could be applied to slightly conducting suspensions. Surface properties ofmonodisperse latex particles (0 9.67 gm), carrying weak acid groups, were characterized by suspension conductometrictitration. This procedure determined the amount of carboxyl groups and showed that strong acid groups were missing on thesurface of these particles. To obtain the electrophoretic mobility, the spheres were separated by free-flow electrophoresis,and the c-potential was calculated from these data. Single-particle rotation experiments on fractions of specified electro-phoretic mobility were carried out at frequencies between 25 Hz and 20 MHz. By analyzing the pH dependence of the rotationvelocity, it could be shown that the rotation rate is determined by surface charges, both at the peak in rotation rate near theMaxwell-Wagner frequency (MWF) and at low frequencies. The inversion of the rotation direction at the MWF peak forvanishing surface charges was demonstrated. An analytical model for the double layer and dissociation on a charged surfacewas developed that is valid for low and high c-potentials. This model could provide convincing evidence of the lineardependence of the MWF rotation velocity on surface charge.

INTRODUCTION

The idea of evaluating the dielectric properties of matter bymeasuring the torque on spheres in circular electric fieldsoriginated in the last century (Arno, 1896). In 1921, P.Lertes measured the torque on filled glass spheres in arotating high-frequency field and determined the dielectriclosses of fluids as predicted by Max Born in 1920. Theusefulness of this method for the measurement of the per-mittivity of highly conducting substances was demonstratedat about the same time (Furth, 1924). This technique can beused as well to assess dielectric and electric properties ofmicroscopic objects like cells and colloids (Arnold andZimmermann, 1982). Measurements can be made by ob-serving the rotation of a single particle, but it is also possibleto use electrorotational light scattering (Gimsa et al., 1995;Priiger et al., 1997) to measure bulk properties in suspen-sions. A technique related to electrorotation is dielectro-phoresis (Kaler and Jones, 1990), which makes use of thetranslational force on particles in AC gradient fields. To-gether, both methods provide a useful tool for the nonde-structive determination of dielectric and electric propertiesof cells and their suspensions (for a review see Foster et al.,1992).

Received for publication 22 April 1996 and in final form 13 June 1997.Address reprint requests to Dr. Hannes Maier, Section of PhysiologicalAcoustics, Department of Otolaryngology, University of Tubingen,Silcherstrasse 5, D-72076 Tiubingen, Germany. Tel.: 49-7071-29-84158;Fax: 49-7071-29-4174; E-mail: hannes.maier@uni.tuebingen.de.C 1997 by the Biophysical Society0006-3495/97/09/1617/10 $2.00

Direct use can also be made of the mechanical torque,translational, and attractive forces to construct devices forthe nondestructive manipulation of cells. Detailed knowl-edge of the underlying forces in electric fields and theirdependencies will improve the construction of microsys-tems that handle and move single cells (Schnelle et al.,1993; Fuhr et al., 1994) or separate cell suspensions accord-ing to their electrical properties (Becker et al., 1995).When the dielectric permittivity and conductivity of latex

particle suspensions were measured, a low-frequency relax-ation at 0.5 kHz (0 1.17 Am particles) was found (Schwanet al., 1962). This could be explained by a diffusion-con-trolled relaxation within a two-dimensional surface layerwith enhanced surface conductivity (Schwarz, 1962). Theimportance of this relaxation for electrorotation is empha-sized by theoretical approaches (Fuhr, 1985; Sauer andSchlogl, 1985; Arnold et al., 1987) in the low-frequencyrange (<10 kHz) as well as for higher frequencies aroundthe peak in rotation rate near the Maxwell-Wagner fre-quency (MWF). Although the surface conductivity seems tobe naturally related to the ionic double layer, there is rareexperimental evidence of a direct relationship (e.g.. Arnoldet al., 1987); there does not exist a clear interrelation be-tween c-potential and surface conductivity. Especially atlow frequencies and conductivity, the two-dimensional de-scription of the surrounding layer may not be sufficient toexplain electrorotation at these frequencies. Volume forceson the counterion layer may contribute to the total force ona particle (Dukhin and Shilov, 1974; Chew and Sen, 1982a;Fixman, 1983).

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Volume 73 September 1997

The present study addresses how electrorotation spectradepend on surface charge and (-potential.

MATERIALS AND METHODS

The actual experiments were carried out in free media far away fromsurfaces. Measurements were not performed on a microscopic slide glass,as was done in similar experiments (e.g., Arnold and Zimmermann, 1982;Fuhr, 1985). This procedure also has the advantage that the homogeneity ofthe electric field and the boundary conditions for the steady-state hydro-dynamic problem of a rotating sphere are well defined (Kirchhoff, 1883).

All rotation experiments were made in a symmetrical three-electrodearrangement. The measuring chamber consisted of three planar, polishedplatinum electrodes of 3 mm x 2 mm surface area (Fig. 1 a). Each of themwas adjusted within a distance of 1.25 mm to the point of symmetry (Fig.1 b). For visual observation, the chamber was closed airtight at its top andbottom with quartz windows. Each electrode was equipped with a phase-and amplitude-measuring device mounted together with the end-stagehybrid amplifiers and the measuring chamber itself on a printed circuitboard. The sinusoidal input signals were supplied by three Hewlett-Packardgenerators (HP-3336C) and were automatically adjusted to the desired ACfield in the volume. With this setup either gradient, homogeneous orrotating electric fields of .100 V/cm could be generated at frequencies of25 Hz to 20 MHz.

h=3mm

a K~

b

The board with the measuring chamber was built into a thermostattedhousing mounted on a manipulator to allow a three-dimensional position-ing of the chamber relative to a fixed microscope. The rotating particlesinside the three-electrode chamber were observed through the windows bymeans of a long-range microscope objective.

An analog computer connected to the three-dimensional positioningdevice was used to indicate when the actually focused region inside themeasuring chamber leaves the internal ellipsoid (Fig. 1 c), to ensure thatonly measurements with a specified error in electric field strength (AE2 <5%; Eq. 2) were taken into account.

At low frequencies electrode polarization occurs. Therefore impedancesof similar Pt electrodes were investigated in electrolyte solutions coveringthe relevant concentration range. The electric field strength was correctedby the frequency-dependent electrode polarization impedance, according tothe phenomenological equation of Fricke (1932) and Schwan (1967), whofound power functions of frequency for Rp and Cp. The factors a, a', b, andb' were determined experimentally with an electrode distance varyingmethod (see Blum et al., 1995):

) b I b'

R = a. t =a'. (1)

((Or is an appropriate, arbitrarily chosen reference frequency for the low-frequency polarization (50 Hz here)).

'V

I'

1i

N

CFIGURE 1 (a) Scheme of the rotating field chamber (h = 3.0 mm). Electrodes were made of massive platinum blocks with planar 2.0 x 3.0 mm surfacesexposed to the electrolyte solution. Microscopic observation was performed through two quartz windows sealed with silicone rubber to the surfaces aboveand under the electrodes. (b) Top view of the electrode arrangement along the central axis. The electrode surfaces have r = 1.25 mm distance to the verticalaxis of symmetry, and the volume has a R = 1.6 mm radius. (c) Time-invariant volume between the electrodes, where relative inhomogeneities AE2 aresmaller than 5% (see Eq. 2). The enclosed ellipsoid shows the volume in which all measurements took place. The electrostatic potential distributions werecalculated numerically for possible potential states of the electrodes during one cycle, assuming that the front windows (top and bottom) were at a distanceof 0.5 mm to the electrodes, which is the worst case possible.

t4 *- '- '- t F--- ..4 .. ...~__,----X ........

1618 Biophysical Journal

....

Surface Charge Dependence of Electrorotation

To determine the actual field strength distribution and the region ofhomogeneity of the electric field in the chamber, a finite-element calculuswas performed. To achieve the three-dimensional potential distribution, thegeometry was covered with a 55 X 55 X 40 discretization net whilesymmetry relationships were taken into account (symmetry along the x-y

plane and the z-y plane, Fig. c). Electrostatic potential distributions forphases from0° to 120° were calculated in steps of5°. With these results thequadratic error relative to the central electric field vector was determinedaccording to the following equation:

AE2 =|E(X, y, z)- Eo(x, y, z = 0)12

JEo(x, y, z 0)12

Fig. c shows the inner volume in which the relative error of the quadraticfield strength does not exceed 5%, and the enclosed ellipsoid shows theanalog-computer controlled volume in which all measurements were made.

For the measurements a sphere in this 5% region was focused, then thefield was switched on and the rotating sphere was tracked until it reachedthe boundary of the 5% region. The whole procedure took between10 s andsome minutes for a particle. The individual rotation velocities were deter-mined off-line by analyzing the video sequences. The time between dif-ferent orientations was stopped several times, from which the individualstandard deviation of the rotation rate was calculated. Rotation of theparticles could easily be detected by eye, because nearly all polystyrenebeads supplied by Polyscience Ltd. showed small inhomogeneities. Diam-eters of the spheres were checked by their video image; those differing bymore than 5% from the mean diameter were rejected.

All chemicals used were of analytical grade. Water was distilled twicebefore it was degassed in vacuum and was kept under nitrogen atmosphereuntil use. Latex particles (Polybeads carboxylatedTM, mean diameter 9.67,um, SD= 0.49,um) were supplied by Polyscience (St. Goar); these had a

surface charge density characterized by the manufacturer as 0.12 meq

COOH/g. Although the beads were supplied as detergent-free, it was

nevertheless advisable to wash them extensively to remove traces ofchemicals that might be carried over from the manufacturing process. Theprocedure was as follows. A particle suspension containing % sphereswas distributed among several rinsed Eppendorf cuvettes, spun down, andresuspended by vortexing a couple of times. Finally, the suspension was

dialyzed against water for 2 days to remove traces of free ions diffusingfrom deeper regions of the spheres. After this treatment conductivity in the

surrounding media remained constant.

Monodisperse latices normally possess additional sulfate groups that

originate from chemicals used in the polymerization process, e.g., initiators

(e.g., persulfate) and copolymers (Fischer and Nolken, 1988). The nominal

surface charge density given by the manufacturer (0.12 meq COOH/g)gives neither information about the solution-accessible part of COOH

groups in the surface layer, nor the quantity of sulfate groups contributing

to the surface charge. Furthermore, changes of the apparent surface charge

density by aging or by oxidation of hydroxyl groups (Shirahama and

Suzawa, 1984) must be taken into account. For this reason, a detailed

analysis of the chemical composition and structure of the surface by

conductometric titration according to the method of Labib and Robertson

(1980) was necessary.

This procedure required all of the particles ( 0.5 g) equilibrated in a 5mmol HCI solution for 24 h. After washing as described above, particles

were immersed in 40 ml water. This suspension was subject to a conduc-

tometric titration with HCI and NaOH to determine the absolute amount of

carboxyl and residual sulfate groups at the surface. A syringe in combi-

nation with a perfusor (no. 1831; Braun, Melsungen, Germany) was used

to add HCl or NaOH (10 mmol both) at a constant rate to a steered,

thermostatted vessel under a nitrogen atmosphere. Conductivity was mea-

sured with a conductivity-measuring probe (WTW, Weilheim, Germany)

and an impedance analyzer (Hewlett Packard 4192A) at 1.592 kHz. The pH

was monitored with a measuring chain connected to a pH meter (Knick,

Berlin, Germany). Measured values were digitized and sampled on a

personal computer. Raw conductivity data had to be corrected for the

constant efflux of electrolyte through the reference electrode diaphragm of

the pH-measuring chain. Further correction was made for the volumeincreases caused by the addition of NaOH and HCl solutions. The remain-ing salt-contaminated suspension was concentrated by centrifugation, dia-lyzed against water, and washed again in the above-described procedure.

Preliminary electrorotation experiments yielded a broad range of rota-tion rates at all field frequencies, although all other parameters were keptconstant. To carry out the electrorotation experiments on fractions withwell-defined f-potentials (with reduced variation), it was thought necessaryto separate particles according to their surface charge on a free-flowelectrophoresis apparatus (ElphorVaP-5; Bender and Hobein, Munich,Germany). The electrophoresis buffer (Hannig and Heidrich, 1990) con-tained 2.2 g triethanolamine, 0.4 g potassium acetate, 18.8 g glycine, and2.0g glucose per liter of solution, adjusted to pH 7.2 with acetic acid.Obtained fractions were washed as described above. Suspensions for allrotation experiments were made of these specified fractions diluted in 0.4mol/liter sucrose solution for density compensation. Particle concentrationswere chosen between 3 mg and 10 mg in 20 ml, except for the pH variationexperiment (Fig. 5), where smaller quantities were used. Conductivity andpH of the different solutions used in the described experiments wereadjusted with HCI, NaOH, and KCI solutions containing 0.4 mol/litersucrose. Conductivity and pH measurements throughout the electrorotationexperiments were done according to the following protocol: 1) Samples ofthe suspension were drawn at the inlet of the measuring chamber beforeand after each experiment. After the particles were removed from the bulksolution by filtering, negligible quantities of indicator para-nitro-phenol(PNP) (10-6mol/l) or methyl red (5.0x 10-7 mol/liter) was added and thepH was checked photometrically. 2) The volume in the system ( 0.25 ml)was replaced every 5-10 min, and the conductivity was checked with aconductivity measuring cell placed in the outlet tube near the rotationchamber. 3) If a sufficient quantity of solution was collected in a syringeat the outlet, the pH was checked as described before.

RESULTS

Conductometric titration

The total number of charged groups as determined by theconductometric titration was 6.9 ,ueq/g polymer. This ismuch lower than the nominal charge density of 0.12meq/g,showing that the particle core is not accessible by themedium. The quantity of dissociable groups leads to a totalsurface charge density of 1.17 neq/cm2. An upper boundaryof the number of sulfate groups could be estimated to be-0.097 neq/cm2. Taking this very conservative value, theratio of sulfate group density with respect to the COOHsurface density of 1.07 neq/cm2 is, at maximum, 10% oreven less.

Electrophoretic mobility distribution

When the 9.67-,tm Polybead COOH particles were sepa-rated by free-flow electrophoresis, the electrophoretic mo-bility showed a very broad distribution around a mean valueof 4.81 X 10-4 cm2/Vs with a standard deviation (SD) of0.22 X 10-4 cm2/Vs (Fig. 2 a). The great variability mayhave its origin in inhomogeneities of the manufacturingprocess, but it is also possible that aging, especially withpretreatment, plays a role, as mentioned by other authors(Bangs, 1984). As expected, the effect on the observedrotation speed was pronounced and decreased the standarddeviation of the rotation rate to nearly one-third for a givenfraction in comparison to the initial composition. The ob-

1619Maier

Volume 73 September 1997

00

0

_t

z

%I-

0

-

0

0

E

z

60

a I501-

40

30

.........i..., ....

20..10~~~~~~~~~~~~~, .... ..,...

_~~~~~~~~~~~~~~. .. ... ...oU~~~~~~~~~~~~~~~._.I .... .....| .

101

3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0

Electrophoretic Mobility [ 1 0-4 cm2/Vs]

70

1.8

( Q Qmean ) / nmeon

FIGURE 2 (a) Distribution of electrophoretic mobility of Polybeads-COOH (09.67 ,um) determined by free-flow electrophoresis. The mean

value of the distribution is 4.81 X 10-4 cm2NVs, with a standard deviationof 0.22 x 10-4 cm2/Vs (pH = 7.2, 40 V/cm, 19°C). (b) Distribution ofangular velocity of rotation fl(v) relative to their mean values fmean(V) ofall data points at the same electric field frequency v. Data shown are forelectric field frequencies v > 20 kHz of a preliminary experiment and theexperiment shown in Fig. 4 a. Compared to the preliminary experimentwith unseparated spheres (SD = 0.179), the variation of the rotationvelocity was drastically reduced (SD = 0.069) if particles of a singleelectrophoretic mobility fraction were used in the electrorotation experiment.

tained variability should be even smaller than that shown inFig. 2 b, because the relative mean values of the angularrotation velocities f1(v)/f1mean(v) were calculated on thebasis of only a few data points per frequency (v, frequencyof the external rotating electric field). This behavior empha-sizes once more the importance of surface charge and C-po-tential for electrorotation experiments.

Electrorotation spectra

In the following experiments the influence of surface chargeon the rotation spectra (rotation rate versus frequency of the

external rotating electric field) was investigated. Two dif-ferent solutions were prepared with 7-mg spheres of fraction28 (electrophoretic mobility (EM) = (4.86 ± 0.12) X 10-4cm2NVs) suspended in 20 ml sucrose solution adjusted to 15AS/cm (one with NaOH, the other with HCl). Furthermore,a second set of experiments was done with 5-mg particles offraction 30 (EM = (4.62 ± 0.12) X 10-4 cm2NVs), but insucrose solutions adjusted to -22 ,S/cm in the same way.Data for the four suspensions are listed in Table 1. Althoughthe particles had been stored in distilled water, the additionto the solutions led to a distinct shift in conductivity. Thishas its origin in the fact that the number of surface-boundgroups is on the same order of magnitude as the totalconcentration of ions in the solutions. Particularly at lowpH, the shift is quite pronounced, indicating that the surfacegroups in the stored fraction were mainly dissociated. Afterthe measuring chamber was rinsed with adequate solutions,particle rotation was observed at various field frequencies inthe range between 50 Hz and 20 MHz. Electrorotationspectra for the four solutions are shown in Figs. 3 and 4,using the dimensionless quantity

2'rfl(v)Nr= E2 (3)

where q is the viscosity of the medium, and fl(v) is theangular rotation velocity in an external rotating electric fieldof electric field strength E (amplitude) and frequency v. TheMaxwell-Wagner frequency (vo) is defined by

2TvO= (oo = (4)

oa, El denote, respectively, the conductivity and the relativepermittivity of the surrounding medium 1 (liquid) or withindex s (solid) of the spheres. E0 denotes the permittivity ofthe free space.The relative dielectric constant of the 0.4 mol/liter su-

crose solution is E, = 74.5 + 1 at 27°C (Landolt andBornstein, 1959), which was the normal cell temperatureduring the experiments.Whereas the rotation spectra at high pH shows a behavior

as described earlier by Arnold et al. (1987) for comparableparticles, the spectra for low pH and vanishing surfacecharge shows a qualitatively different behavior. In the caseof low pH, the peak in rotation rate near the MWF inverts,with a clearly pronounced minimum at this point (Fig. 4 b).To our knowledge this experiment is the first observation ofa rotation in the direction opposite that of the excitingelectric field for vanishing surface charge.

Surface charge dependency of the rotationvelocity at the MWF

The results from the preceding experiment made it neces-sary to examine the effects of surface charge and pH indetail. For this purpose it is essential to restrict pH andconductivity to a defined range. Attempts to use various

/V I I I I I I

1620 Biophysical Journal

7::::::7

'o k.I I

:::.?I :::::i:

:::: tI

Surface Charge Dependence of Electrorotation

TABLE 1 Electrophoretic mobility and pH of the suspensions (0.4 mol/liter sucrose) determined during the electrorotationexperiment shown in Figs. 3 and 4

No. Fraction no. u [10 4] (cm2 Vs) Umean (jtS/cm) MWF (kHz) Temp. (°C) pH

1 28 4.86±0.12 11.78+0.12 284+7 27.0+0.1 5.38+0.152 28 4.86 + 0.12 11.70 + 0.57 274 + 17 26.8 + 0.1 7.47 + 0.13 30 4.62 + 0.12 14.94 + 2.47 364 + 64 27.0 + 0.1 4.13 + 0.304 30 4.62 ± 0.12 22.07 + 0.36 536 + 11 27.0 ± 0.1 8.30 ± 0.12

buffers were unsuccessful. We therefore used mixtures ofNaOH and acetic acid without any stabilization of the pH,but with a rigorous control of pH and conductivity. Theconductivity was chosen to be -22 ,S/cm (MWF vo 500

kHz), and 10 mixtures in the pH range accessible underthese conditions were prepared (see Table 2). Quantities ofless than 1 mg of particles of fraction 24 (EM = (5.33 ±

0.12) X 10-4 cm2/Vs) were added to 5 ml of each solution.

z

z

FIGU

(09.6in a (obtainobtaindeviat

The negligible amounts of spheres gave rise to only a minoralteration of pH when compared with the photometricallyobtained values at the outlet of the measuring chamber(before and after the experiment). Volume and tubings were

rinsed with -200 ml of the identical solutions before thesystem was filled with the suspension. Results of the ob-served rotation velocities at the MWF are plotted in Fig. 5.

DISCUSSION

0.15 a Conductometric titrationI: p H=7.5 The "parking area" for the closest packing of COOH groups

on a two-dimensional surface is 20 A2 (Seradyn, Inc.,1992). The surface density of 1.07 neq COOH/cm2 found in

0.1 _ the conductometric titration is equal to a carboxyl groupw I v0=2f4+/ kHlz "parking area" of 15 A2. This smaller value indicates that

I zs the characteristic groups are not only distributed on a two-dimensional surface, but are also buried in a thin three-

0.05 _ , - z dimensional subsurface layer. From the ratio of nominal

COOH density to the apparent density results follows a

..~ -. - I* - thickness of the swollen (water containing) shell of -100nm. Normally core sizes and permeability of water vary in

102 103 104 105 106 107 a broad range, depending on various factors like polymer-ization procedure and number of cross-links (El-Aasser etal., 1985). Our results show that the 9.67-,um Polybeads-

0.05 COOH expose an ion exchanger-like surface layer with al b - negligible number of sulfate groups, which is thin (2%)

0.04 pH= compared to the diameter of the sphere. Because water doesnot seem to exist in the core region of the spheres, thepermittivity and conductivity of the solid can be estimated

0.03 to be similar to that of the pure compound (polystyrene: Dk'I ,o.=284+-17 kHz = 2.5; Saechtling and Zebrowski, 1967). This will not be

true in a swollen boundary layer containing water, where an0.02 _ -= - enhanced permittivity and conductivity must be taken into

consideration. In addition, this boundary region will expose

0.01 ion exchanger-like properties; hence apparent electrical sur-*~ Ii s -I face charge is not necessarily equal to the amount of titrated

(chemical) surface groups. In this case charged groups in the0.0 "2 '';"''"' ''" solid matrix are partially compensated by counterions in the

enclosed liquid phase. For this reason the titration was onlyFrequency [Hz] used to determine the depth and composition of the water-

containing boundary layer, and apparent surface chargesIRE 3 Electrorotation spectra of the carboxylated latex particles and potentials were calculated from the electrophoretici7 ,im, electrophoretic mobility u = (4.86 ± 0.12) x 10-4 cm2/Ns) mobility.).4 mol/liter sucrose solution. (a) With bulk pH = (7.47 + 0.1), . .ied by the addition of NaOH. (b) With bulk pH = (5.38 ± 0.15), In addition, the conductometric titration showed an es-ied by the addition of HCl. The mean MWF v. and its standard sential feature of our particles: the acid groups contributingtion during the experiment are indicated by vertical bars. to the surface charge are mainly, if not exclusively, carboxyl

1621Maier

1622 Biophysical Journal ~~~~~~~~~~Volume73 September 1997

I ~~~~~apH=8.3*!1

I

I:I

v.=536+-ll kHz

Electrophoretic experimentsBy separating the particles according to their electrophoreticmobilities at a given pH, the effective "electrical surfacecharge " was determined. Using the Helmholtz equation,

(5)606i

0.05 * -the mean electrophoretic mobility u taken over all fractions

[ ~~~~~~~~~~~~190C).Unfortunately, at a pH of the separation medium of

0.0.~~~~~~~~~~~ ~7.2, the weak acid groups were not fully dissociated, as

'2-~~~~~~~~~ ~could be seen later in the electrorotation experiments. Cor-102 0Io 0' io5 106 107 recting this value according to the model function in Fig. 5,

a c-potential for completely dissociated carboxyl groups of-84.5 mV results.

..I ~~~~~~~~~~~~Ascan be seen in Fig. 2b, the restriction toparticles of0.03 -k a smaller range in electrophoretic mobility than in the initial

I US H= .1distribution has a pronounced effect on the variation inP H 4 rotation velocities at field frequencies around the MWff.

0.02 -This reduction can be interpreted as a first hint of therelative importance of surface potentials and surface

0.1xI v0=364+-64 kHz charges for the rotation in electric fields.

I ZI -E0.0 ........................ Electrorotation spectra

411 ~~~The results at high pH can be compared with results ob--0.01 tamned in earlier experiments (Arnold et al., 1987). This

2 *,,,~~~~I I I~~~publication includes measurements of particles that are di-10, lo 1 0 410 , I 0e 1 0 rectly comparable to those used in our experiments. Their

Frequency[Hz]~~particles were supplied by the same manufacturer, and were

Frequency[Hz] ~~of the same diameter and standard deviation, so that they

RE 4 Electrorotation spectra of the carboxylated particles (09.67 prblyeontohesmbac.-Petisasesud

-lectrophoretic mobility u = (4.62 +0.12) X 10-4 cm2NVs) in 0.4 by the former authors (-84 mV) appear to be comparable toiter sucrose solution. (a) With pH =(8.30 ± 0.12), obtained by the our corrected values. In both investigations, the linear de-on of NaGH. The mean conductivity was (22.07 ± 0.36) pS/cm. The pendence of rotation velocity on quadlratic field strength as

~dcurve is a least mean square fit of the model without surface well as the decrease in rotation velocity at its specific MWVFes (Eq. 21) to the high-frequency part (>10 kHz) of the electrorota- with increasing ionic concentrations is confirmed. When-pectra. The ratio of the dielectric constants for the sphere and theim was assumed to be U =2.5/74.5, and the fitting procedure leads comparing our electrorotation spectra at high pH with thoseconductivity ratio of K =0.0153. (b) With pH = (4.13 ± 0.3), obtained by Arnold et al. (1987), a qualitative and quanti-ied by the addition of HCI. The mean conductivity was (14.94 ± tative agreement was apparent. The two maxima were

tLS/cm. The dashed curve is a fit of the surface charge incorporating found: at or above the MWF and at low frequencies (<101 (Eq. 23) to the high-frequency part (v> 10 kHz) of the data, leading kHz). Taking our measurement where the conditions are=9620 and p = 59600 for U = 2.5/74.5. The temperature was comparable (Fig. 3 a), the ratio between the peak velocities

0rdevitio during bthe experimentsarei endiae byFvertialdbars. at MWF and at 100 Hz is -~1.9, indicating that the twoarddevaton urng he xprimntareinicaed y ertca bas.

systems are in a similar state of dissociation. To comparethe values of rotational velocity, corrections must be made

ps. Generally, strong acid groups (sulfate groups) orig- for the differing viscosities of water (210C) and the sucroserng, e.g., from persulfate (which is used as an initiator in solution (270C) (qlk,2 = 0.978 cPoise/1 .533 cPoise). Fur-polymerization procedure), are present on nearly all ther compensation must then be made for the differentsof such particles. The absence of strong acid groups conductivities and MWF of the two solutions. Taking thei)ur particles makes it possible to suppress surface model of Arnold et al. (1987) (Eq. 1 1), direct comparison ofges nearly completely in the medium pH range. The the peak rotational velocities at the MWF can be made.~ion of strong acid groups would have a pH too low (and Using the parameters 1) sucrose: 11.8 pxS/cm, Dk = 74.5;niductivity too high) required to reach an undissociated and 2) water: 7 ViS/cm, Dk = 79, the peak rotational

velocities were equal within 5-10%.

z-

L.

FIGUpim,emol/liiadditi(dashe(chargition s]mediuto aobtain2.47),modelto A(27.0standa

groulinatiithetype,~on (charltitrat,a coistate,

.....I . . ....I I I . ....I . ....I . ..'.I .

1622 Biophysical Journal

'r

0.1 . -

Surface Charge Dependence of Electrorotation

TABLE 2 Initial solutions (0.4 mol/liter sucrose) before the spheres were added (u = (5.33 ± 0.12) x 10-4 cm2Ns) and particlesuspensions used in the pH variation experiment

Experiment no. Solution pH Solution a (,S/cm) Suspension pH Suspension o- (,AS/cm) MWF (kHz) Temp. (°C)

1 9.11 21.7 8.84 22.31 ± 0.06 538 ± 10 26.82 7.90 22.1 7.14 23.48 ± 0.07 567 ± 10 26.93 6.99 21.2 6.71 22.46 ± 0.05 542 ± 10 27.04 6.46 21.4 6.41 22.95 ± 0.04 554 ± 10 27.05 5.97 21.2 6.18 22.69 ± 0.12 547 ± 10 27.16 5.75 21.7 5.82 22.99 ± 0.05 555 + 10 27.17 5.53 21.3 5.64 22.36 ± 0.04 540 ± 10 27.28 5.00 21.5 5.15 22.55 ± 0.09 544 ± 10 27.29 4.50 21.4 4.49 22.96 ± 0.11 554 ± 10 27.210 4.05 21.6 3.94 21.08 ± 0.14 509 + 10 27.2

Suspension parameters were determined throughout the experiment.

To quantify the pronounced dependence of the rotationrate at the MWF on surface charge, the following model wassuggested. According to the mass balance equation, thedissociation a = CH+/CCOOH for a given concentration ofprotons is given by

aK = CH, I - a (6)

potential at a distance r from the center of the sphere isgiven by the solution of the one-dimensional Poisson-Boltz-mann equation:

d =

2deC=C-C+ (7)

with the dimensionless variables

For a charged surface the concentration of charges at thesurface and in the Gouy-Chapman double layer are notequal to the bulk concentrations. In case of a small Debyelength SD (small compared to the radius A of the sphere), the

0.04

0.03

-'-N 0.02

Z 0.01

0.0

-0.01

**** ' TA

... . . . . . .~~~ -s ,3/S~~~~~~~~~~~~~X~~~~~

laAS.. I/

_~ ~ ~~~,- /

la/

I I3.0 4.0 5.0 6.0 7.0 8.0 9.0

pH

FIGURE 5 Dependence of the rotation velocity at the MWF (vo = 500kHz) on the pH of the surrounding medium (0.4 mol/liter sucrose). Car-boxylated latex particles (0 9.67 ,tm, electrophoretic mobility u = (5.33 +0.12) X 10-4 cm2/Vs at pH = 7.2) having a maximal f-potential of -93.6mV according to the Helmholtz equation (Eq. 5) when fully dissociated.Conductivity was kept constant (22.58 ± 0.43 ,tS/cm) while the pH wasvaried between 3.94 and 8.84. The "best fit" procedure of the model in Eq.22, assuming a surface charge of F = 11.87, leads to pK 4.91 with a lowerlimit of N? = -0.0103 and a span of ANr = 0.0502 for Nr(Vo). Forcomparison, the "normal" symmetrical titration behavior (Henderson-Has-selbalch) for pK 4.91 is shown, taking no surface charges into account(dashed line). Measurements were performed at v = 500 kHz field fre-quency and a field strength of -100 V/cm.

pF c+ r-ART C_cD

(8)

where (p is the electrical potential; c+, c_ are the ionconcentrations; and co is the bulk concentration of the ions(assuming a 1-1 electrolyte and electroneutrality at infin-ity). F denotes the Faraday constant, R the gas constant, andT the absolute temperature. Ion concentrations are given bythe electrochemical potential equilibrium:

ln C_ ± '1 = const. (9)

leading with r -o c => C+ -> 1 to

2= sinh D (10)

The solution of this differential equation is given by Sher-wood (1979 (unpublished Ph.D. thesis cited in Chew andSen, 1982b)):

= + te-) 2

(D = In 1- te-6 (11)

with

t = tanh((1F) 2 eIRTD 2F2C0 (12)

where (D is the dimensionless electrical potential at thesurface (C-potential). This result is valid for all (D and leadsin the approximation for t < 1 to the well-known Gouy-Chapman equation. Substituting the proton concentrationgiven by Eqs. 9 and 11 and using the dimensionless con-

U.UJ)

-U.UL'

1623Maier

_n

Volume 73 September 1997

centration at infinity C(, Schlogl (1985) for a noncharged sphere,

-(1 te ~2= C~2e~"~ = I~ ± te-

(13)

In Eq. 6 a modified Henderson-Hasselbalch equation on acharged surface results:

pH = pK + log (1t)2) (14)

The additional term incorporates the influence of a surfacepotential on the dissociation of surface groups. The rela-tionship between the surface charge and the surface poten-tial is given by the first Maxwell equation, V * D =c,yielding with the surface charge y (mol/cm2),

ap Fy (15)ar=-F

and in its dimensionless form for one dimension,

&(F F&( 2 (16)

with the dimensionless surface charge F = 'YCO8D.Solving this differential equation, one obtains

t2+8tt2+ -1=0 (17)F

and in the case of negative surface charges (F < 0), theresults

4 / /4X2 1/2

t= -r - I + -v)) (18)

Now substituting t in Eq. 14, an algebraic equation offourth-order results, giving the relationship between disso-ciation a and pH for a given surface concentration ofdissociable groups:

a2(1 + b) 2 a2(1+ b) a2b(X a 4b a2 t + =+ (19)aa+ ~~4b 2 4

with

b 10(pH-pK) (20)

This equation may be solved analytically by the method ofL. Ferrari (1522-1565), but it is more comfortable to solveit numerically with the Newton method. Introducing Eq. 17in Eq. 14 leads to an inverse function pH(a) with a singlesolution, which allows us to distinguish between severalpossible solutions of Eq. 19 by choosing PH - pH(a(pH))I< E as a criterion for convergence to the "right" solution.From a "best-fit" procedure of the model of Sauer and

3(K-U) XIXr(K + 2)(U + 2)1 +(XXr) (21)

with

's; uBs = v K+ 2

K=-; U= ; X= ; rU+O', 61 VO U+2

to the low pH data (Fig. 4 b), it follows that the conductivityof the solid is small compared to that of the liquid. Thismust not be true in general, especially in a water-containing"swollen" state with fixed surface charges. Using the modelof Arnold et al. (1987) leads to a similar result. Under thecondition of low conducting spheres, the approximation ofthe model used in this source results in a linear dependenceof rotation on surface conductivity. With this working hy-pothesis in mind, the following "best fit" procedure of alinear dependence of rotational velocity (at the MWF vo) onsurface charge was applied:

Nr(vo) = No + ANr- a(pH) (22)

The surface potential of fraction 24 used in this experimentwas calculated from the electrophoretic mobility with theHelmholtz equation (Eq. 5). It yields ; = -81.9 mV, F =

9.32, and ; = -93.6 mV, F = 11.87 if the correction forpartially dissociated surface groups were taken into account.The result of the curve fit is shown as the solid line in Fig.5, showing an excellent approximation of the experimentaldata. It results further in an independent pK of 4.91 (; =-93.6 mV) for the carboxyl groups. This value is muchmore realistic with respect to carboxyl groups in this kind ofchemical binding than the value of the half-maximum ef-fect, which is -6.0 (for comparison: benzoic acid: pK =

4.22; 2-phenyl-propionic acid: pK = 4.64;CH3(CH2)nCOOH: pK = 4.82-4.95 (n = 2-7), which areall below a pK of 5). In addition, the "normal" symmetricaltitration behavior in the absence of a surface potential is notable to approximate the much broader and asymmetricaldependence on the bulk pH. For comparison, this normaltitration behavior of surface groups in the absence of asurface potential is shown as a dashed line in Fig. 5, assum-ing a pK = 4.91. From this model it becomes apparent thatthe assumption of a linear dependence of rotational velocityon surface charge is adequate to describe the rotationalvelocity at the MWF maximum. The model function im-plies, furthermore, that the lowest investigated pH still leadsto a dissociation of -10%. To estimate the residual rota-tional velocity at low frequencies, the ratios between max-imum (at high pH) and residual mean velocity (at low pH)at the five lowest frequencies (50-200 Hz) were calculated(Fig. 4, a and b). The residual rotational velocity at theselow frequencies is 13% on average, with a standard devia-tion of 2%. In the case of a linear dependence, this corre-

spondence indicates that the low-frequency rotation ismostly or totally due to the surface charge system. This

1 624 Biophysical Journal

Maier Surface Charge Dependence of Electrorotation 1625

analysis shows that the models of Sauer and Schlogl (1985)

3X(PX2(A - PU) - A - Ur X2(X + 2 + A + 2P)2 + (2- (U + 2)PX2)2 (23)

with

Es AE vU A= ; X=-; P = 21rV0Ta81 ElO

hold for the presence and absence of surface charges (seealso Fig. 4, a and b) around the MWF. It is also probablethat this model describes the behavior at low frequencies fornoncharged particles, but does not hold for surface chargedparticles in this frequency range.

CONCLUSION

The behavior of colloids in electrorotation experiments isdominated by the presence of surface charges. This includesthe high-frequency range around the MWF as well as thelow-frequency region (<10 kHz). The high-frequency(MWF) rotation may be approximated by a surface-charge-dependent linear term. A model for surface-bound chemicalgroups incorporating the surface potential was developedthat describes the experimental data in an adequate manner.In the case where the conductivity of the particle is smallcompared to that of the surrounding liquid, it was shownthat the effect may be so drastic that it can invert the directionof the rotation. The rotation rate at low frequencies is mainly orperhaps totally determined by the surface charge system. Theinfluence of the surface charge may be described in the high-frequency domain either directly or as an enhanced surfaceconductivity in the sense of G. Schwarz. This need not to betrue for the low-frequency region.The importance of surface charge systems may be em-

phasized not only for solid particles but also for multishellsystems as spherical cells. A further implication arises fromthe Kramers-Kronig integral relationship, so that major in-fluences can also be expected in particle-particle interac-tions as well as in dielectrophoretic experiments.

I am grateful to Prof. F. A. Sauer for the support of the underlying doctoralthesis and for his help with the theory and many helpful discussions. I amalso obliged to Prof. K. Hannig for quickly supplying a free-flow electro-phoresis apparatus when needed. Dr. J. P. Fischer of Hoechst AG was kindenough to introduce me to the conductometric titration technique of col-loids. I thank Prof. F. Eicke, Dr. C. P. Richter, and Dr. A. Rusch for theirvaluable comments on an earlier version of this paper. This work is part ofa doctoral thesis carried out and supported by the Max-Planck-Institut furBiophysik, Frankfurt am Main.

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